Playing with Numbers

Claire Burrin is fascinated by numbers. Not fractions, probabilities or differential equations – just regular whole numbers. In mathematics, these are the domain of number theory.

Burrin is in good company here, as numbers have been a topic of theorizing since ancient times. Number theory, along with geometry, is by far the oldest branch of mathematics. We know of nearly 4,000 clay tablets from ancient Babylonia containing tables of reciprocals, products and combinations of whole numbers. Aristotle was already puzzling over prime numbers as far back as the 4th century BC – numbers that cannot be divided evenly by any other number, such as 2, 3, 7, 11, 17, 19, 23, 37, et cetera.  “Prime numbers are essentially the atoms of mathematics,” says Burrin. “They are the building blocks on which all other numbers are based.”

But isn’t fiddling around with numbers a dry and boring task? Professor Burrin returns my question with a pensive gaze. “Let me give you an example,” she says. “Take any number, let’s say the number 10. If the number is even, we divide it by two, which gives us 5. But if it’s odd like 5, we multiply it by 3 and add 1. Now we’re at 16. 16 is even and can be divided by 2, giving us 8. We repeat the exercise and then we’re at 2, at 4, and then finally at 1.” So far I’m able to follow. “This pattern always holds,” she says. “We always end up back at 4, 2, 1. No matter what number we begin with.” – “Every number?” I ask. “Always? “Even if we start with 2,357?” Burrin smiles: “See, you’re already curious about the material.”

She caught me. Admittedly, I want to know the answer and don’t find this numbers game boring at all. But it’s much more than just a game. Apparently, some numerical ratios and sequences are so fundamental that they permeate the entire natural world (see box). Take the Fibonacci sequence, for instance, in which each number is the sum of the preceding two figures: 0, 1, 2, 3, 5, 8, 13, 21, 34, 55. These proportions are found in the spiral shells of nautiluses and in the scales of pinecones, pineapples and artichokes.

But why? We know part of the answer. Fibonacci spiral structures in plants allow leaves or seeds to be arranged with minimal overlap, which optimizes their collective use of light, space and nutrients. This simple mathematical relationship ensures both beauty and efficiency.

What’s less beautiful is Burrin’s extremely spartan office. There is barely any furniture and nothing hanging on the walls. Only a few papers and books are stacked haphazardly on her desk and on the one shelf that’s present. The desk is only home to a monitor and a laptop. The first impression seems to confirm the stereotype of mathematicians as people who think in a different realm than ordinary mortals and whose work mainly takes place in their heads.

What does lend the office a sense of energy is the blackboard, which takes up an entire wall and is filled with drawings and notes. Some parts flow into others, and you can see how the sketches and formulas have all built on each other. “I use my blackboard every day,” says Burrin. Not to puzzle over formulas in solitude, but to collaborate with colleagues and team members – to sketch out and discuss ideas together. This actually contradicts the cliché of the ivory tower mathematician who mulls over numbers and theories alone in a quiet room. “On the contrary, math is an extremely cooperative and social field,” says Burrin. Of course, part of her research involves reading mathematical papers and conducting experiments, mostly with the help of numerical computer models. However, she also spends a lot of time exchanging ideas with colleagues and exploring whether her thoughts might remind them of something from a different context.

Earlier this morning, she spent time at her blackboard with a student, working through ideas for a Master’s thesis. She points to the sketch in the lower right corner: a two-dimensional lattice of parallelograms in white, with a circle drawn in yellow chalk at each corner of the shapes. “The question is how we can arrange these circles in the most space-efficient way possible,” she explains. When dealing in two or three dimensions, it’s still possible to sketch out or imagine what this would look like. In three dimensions, the answer is cannonball packing, or creating a pyramid of stacked spheres. “But for most higher dimensions, we still have no idea what the densest possible arrangement would look like,” says Burrin. Although this seems like a highly abstract line of questioning, it has real-world implications. The answer could be used to pack data as densely as possible for electronically transmitting information, for example.

One example of a numerical ratio found throughout nature is the famous golden ratio – a proportion that architects, photographers and designers strive for, knowing that we perceive these proportions to be especially harmonious and beautiful. It’s a fairly simply mathematical relationship: a segment possesses the golden ratio if the ratio of the longer segment to the shorter segment is the same as the ratio of the whole line to the longer segment. Or, expressed as a formula, a/b=(a+b)/a. This simple relationship is ubiquitous in nature. In bees and wasps, for example, the lengths of the thorax and abdomen correspond precisely to these proportions. Or consider the five-petaled flowers of many plant species: their petals form a regular pentagon, which has sides that relate to its diagonals in the golden ratio.

The Fibonacci sequence works in a very similar way, generating proportions that approach the golden ratio without ever quite reaching it. The scales of a pinecone, the seeds of a sunflower, the florets of a cauliflower – all of these naturally occurring patterns are arranged in spirals typically underpinned by consecutive Fibonacci numbers. These two compellingly simply mathematical ratios have independently asserted themselves countless times over the course of evolutionary history.

Concrete applications are not what drives Burrin’s research, however. She is motivated by pure curiosity. For most of her work, she uses lattices to investigate the symmetry properties of surfaces and their relationship to certain numerical invariants – values that stay constant under mathematical operations. “Basically, I combine aspects from geometric theory and number theory to gain a better understanding of these relationships,” says Burrin.

An easy-to-understand example of the relationship between numbers and shapes is the circle: a perfectly symmetrical figure. To measure the radius and the diameter, we need the number π, which begins with 3.14159. Categorized as a transcendental number, pi is infinitely long and complex, as it doesn’t have any known repeating sequences or patterns. While it seems almost inconceivable that such a chaotic number describes such an ideal shape, Burrin finds it to be a thing of wonder and beauty.

“Every mathematical insight begins as a theoretical, perhaps even seemingly naive, question posed by someone who wants to understand the symmetry of a shape or the origin of a pattern,” says the professor. “In this way, abstract ideas can profoundly shape the world and our everyday lives, sometimes even centuries after the fact.”

One example of this is Aristotle’s insight into prime numbers. What makes them fascinating is that multiplying even two very large prime numbers together is a straightforward exercise. Today’s computers manage the task in mere milliseconds. But the reverse – breaking a product back down into its prime factors, a process number theorists call factorization – is much more difficult, especially for very big numbers. After all, numbers can be infinitely large. Once a number reaches several hundred digits, algorithms need an unimaginably long time for factorization, making the task practically impossible. This is precisely why this process is used in modern encryption protocols such as SSL/TLS and SSH. Almost everything we send over the internet, from emails to online banking passwords, is encrypted through these protocols in dimensions of mind-boggling scale.

A new development is that AI is supporting mathematical research in umpteen dimensions. For instance, computer programs can verify proofs and examine complex structures that exceed human capabilities. AI also helps programmers write mathematical algorithms, which is a great boost to efficiency. Burrin also uses AI in her work, but maintains that human creativity is irreplaceable. While humans ask the questions and think about the possible solutions, she says, we still need to question findings and distinguish between reliable knowledge and mere output.

Speaking of reliable knowledge, what about the number game we played at the beginning, where you always keep landing at 4, 2 and 1? Does that really work with every number? No one actually knows. As simple as the calculation is, there is no mathematical proof to back it up. “Algorithms have been deployed to calculate through to very high numbers, and until now it’s worked for all of them,” she says. Nevertheless, as long as there is no mathematical proof, it remains possible that there is a very high number for which the pattern doesn’t hold. But that’s the beauty of mathematics, according to Burrin: seemingly simple puzzles can be difficult to solve, but once a mathematical relationship is proven, it is valid everywhere and for all time – a type of eternal truth.